May 23, 2008

There is a 9 digit number.Its speciality is no digit is repeated.and its right 1 digit is divisible by 1 and right 2 digits is divisible by 2,right 3 digits is divisible by 3.This pattern will be repeated until 9 digits.

Although there are several correct answers, none of you have got it right so far.

The smallest answer is 123567480 and the largest is 875643120.

Since the number formed by the right 2 digits is divisible by 2, the last digit must be even. Since the right 5 digits are div by 5, the last digit must be 5 or 0. The only answer satisfying both these conditions is 0. So the last digit must be 0.

Only Ignophile's first answer fits these criteria - all the others are clearly wrong. However, 381654720 is not correct either because 1654720 div by 7 leaves a remainder of 4.

So, to continue...

The last 8 digits must be div by 8. Only numbers ending in ...00, ...20, ...40, ...60, and ...80 are div by 8 and ...00 is excluded because it repeats a digit.

The last 3 digits are div by 3 and only numbers whose digits sum to a multiple of 3 are div by 3. This gives the following as the only possible last 3 digits:120, 240, 360, 480, 720, 840. Interestingly, all of these will be div by both 4, 5, and 8 no matter what digits are put to the left of them so lets ignore them and move on.

To be div by 9 the digits of a number must sum to a multiple of 9. Adding the 10 digits together (0+1+2+...+8+9) gives 45 which is div by 9. However, this is a 9-digit number so we have to drop one from the ten. We can only take out 0 or 9 and still have the remaining digits sum to a multiple of 9. We've already shown that 0 is present so the 9 is dropped.

Similarly, to be div by 6 a number must be even (it is) and have its digits sum to a multiple of 3 (it does). So the final part is making the last 7 digits div by 7.

No neat rules here so just plug some digits in and swap them around until you get a number that is evenly div by 7 (Excel is good for this).